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Subsections
7.2 Energy gradients
Having calculated the total energy, both the density-kernel
and the expansion coefficients for the localised orbitals
are varied. Because of the non-orthogonality
of the support functions, it is necessary to take note of the tensor
properties of the gradients [163], as noted in section 4.6.
The total energy depends upon
both explicitly and through the
electronic density
. We use the result
![\begin{displaymath}
\frac{\partial K^{ij}}{\partial K^{\alpha \beta}} = \delta_{\alpha}^i
\delta_{\beta}^j .
\end{displaymath}](img966.gif) |
(7.20) |
From equations 7.6, 7.16 and 7.18 we have that
![\begin{displaymath}
E_{\mathrm{kin,ps}} = T_{\mathrm s}^{\mathrm J} + E_{\mathrm...
...NL}} =
2 K^{ij} (T + V_{\mathrm{loc}} + V_{\mathrm {NL}})_{ji}
\end{displaymath}](img967.gif) |
(7.21) |
and therefore
![\begin{displaymath}
\frac{\partial E_{\mathrm{kin,ps}}}
{\partial K^{\alpha \bet...
... 2 (T + V_{\mathrm{loc}} +
V_{\mathrm {NL}})_{\beta \alpha} .
\end{displaymath}](img968.gif) |
(7.22) |
The sum of the Hartree and exchange-correlation energies,
depends only on the density so that
![\begin{displaymath}
\frac{\partial E_{\mathrm{Hxc}}}{\partial K^{\alpha \beta}} ...
...f r})}
\frac{\partial n({\bf r})}{\partial K^{\alpha \beta}} .
\end{displaymath}](img970.gif) |
(7.23) |
The functional derivative of the Hartree-exchange-correlation energy with
respect to the electronic density is simply the sum of the Hartree and
exchange-correlation potentials,
. The
electronic density is given in terms of the density-kernel by
![\begin{displaymath}
n({\bf r}) = 2 \phi_i({\bf r}) K^{ij} \phi_j({\bf r})
\end{displaymath}](img972.gif) |
(7.24) |
so that we obtain
![\begin{displaymath}
\frac{\partial n({\bf r})}{\partial K^{\alpha \beta}} =
2 \phi_{\alpha}({\bf r}) \phi_{\beta}({\bf r}) .
\end{displaymath}](img973.gif) |
(7.25) |
Finally, therefore
![\begin{displaymath}
\frac{\partial E_{\mathrm{Hxc}}}{\partial K^{\alpha \beta}} ...
...r})
\phi_{\alpha}({\bf r}) = 2 V_{\mathrm{Hxc},\beta \alpha} .
\end{displaymath}](img974.gif) |
(7.26) |
Defining the matrix elements of the Kohn-Sham Hamiltonian in the representation
of the support functions by
![\begin{displaymath}
H_{\alpha \beta} = T_{\alpha \beta} + V_{\mathrm{Hxc},\alpha...
...
V_{\mathrm{loc},\alpha \beta} + V_{\mathrm {NL},\alpha \beta}
\end{displaymath}](img975.gif) |
(7.27) |
the derivative of the total energy with respect to the density-kernel is
simply
![\begin{displaymath}
\frac{\partial E}{\partial K^{\alpha \beta}} = 2 H_{\beta \alpha} .
\end{displaymath}](img976.gif) |
(7.28) |
Again we can treat the kinetic and pseudopotential energies together,
and the Hartree and exchange-correlation energies together. We use the
result that
![\begin{displaymath}
\frac{\partial \phi_i({\bf r})}{\partial \phi_{\alpha}({\bf r'})} =
\delta_i^{\alpha} \delta({\bf r}-{\bf r'}) .
\end{displaymath}](img977.gif) |
(7.29) |
We define the kinetic energy operator
, whose matrix elements are
![\begin{displaymath}
T_{ij} = \int {\mathrm d}{\bf r}~ \phi_i({\bf r}) {\hat T} \phi_j({\bf r}) .
\end{displaymath}](img979.gif) |
(7.30) |
Since the operator is Hermitian,
![\begin{displaymath}
\frac{\delta T_{ij}}{\delta \phi_{\alpha}({\bf r})} =
\delta...
...\phi_j({\bf r}) + \delta_j^{\alpha} {\hat T}
\phi_i({\bf r}) .
\end{displaymath}](img980.gif) |
(7.31) |
Therefore
The derivation for the pseudopotential energy is identical with the
replacement of
by the pseudopotential operator, and so the result
for the sum of these energies is just
![\begin{displaymath}
\frac{\delta E_{\mathrm{kin,ps}}}{\delta \phi_{\alpha}({\bf ...
... + {\hat V}_{\mathrm{ps,tot}} \right)
\phi_{\beta}({\bf r}) .
\end{displaymath}](img985.gif) |
(7.33) |
Again this gradient is derived by considering the change in the
electronic density.
![\begin{displaymath}
\frac{\partial n({\bf r'})}{\partial \phi_{\alpha}({\bf r})}...
..._i({\bf r'}) \delta({\bf r} - {\bf r'}) K^{i \alpha} \biggr] .
\end{displaymath}](img986.gif) |
(7.34) |
Therefore
![\begin{displaymath}
\frac{\delta E_{\mathrm{Hxc}}}{\delta \phi_{\alpha}({\bf r})...
...K^{\alpha \beta} {\hat V}_{\mathrm{Hxc}} \phi_{\beta}({\bf r})
\end{displaymath}](img987.gif) |
(7.35) |
The gradient of the total energy with respect to changes in the
support functions is
![\begin{displaymath}
\frac{\delta E}{\delta \phi_{\alpha}({\bf r})} = 4 K^{\alpha \beta}{\hat H}
\phi_{\beta}({\bf r})
\end{displaymath}](img988.gif) |
(7.36) |
where
is the Kohn-Sham Hamiltonian which operates on
.
Next: 7.3 Penalty functional and
Up: 7. Computational implementation
Previous: 7.1 Total energy and
  Contents
Peter Haynes